It is noteworthy that already in this mathematical formulation of experimentally well-confirmed fundamental physics the seed of higher differential cohomology is hidden: Dirac had not only identified the electromagnetic field as a line bundle with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological Chern class with the (physically hypothetical but formally inevitable) magnetic charge located in spacetime. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell’s equations imply that at the support of any magnetic charge the 2-form representing the field strength of the electromagnetic field is in fact not closed and hence in particular not the curvature 2-form of an ordinary connection on an ordinary bundle.

In summary, the experimentally verified models, to date, of fundamental physics are based on the notion of (twisted) U(n)U(n)-principal bundles with connection for the Yang-Mills field and O(d,1)O(d,1)-principal bundles with connection for the description of gravity, hence on nonabelian differential cohomology in degree 2 (possibly with a degree-3 twist).

In attempts to better understand the structure of these two theories and their interrelation, theoretical physicists were led to consider variations and generalizations of them that are known as supergravity and string theory. In these theories the notion of gauge field turns out to generalize: instead of just Lie algebras, Lie groups and connections with values in these, one finds structures called Lie 2-algebras, Lie 2-groups and the gauge fields themselves behave like generalized connections with values in these.

So far all these differential cocycles were known and understood mostly as concrete constructs, without making their abstract home in differential cohomology explicit. It is the next gauge field that made Freed and Hopkins propose (FreedHopkins, Freed) that the theory of differential cohomology is generally the formalism that models gauge fields in physics:

The superstring is charged also under what is called the RR-field, a gauge field modeled by cocycles in differential K-theory. In even degrees we may think of this as a differential cocycle whose curvature form has coefficients in the ∞-Lie algebra⊕n=0∞B2n𝔲(1)\oplus_{n=0}^\infty \mathbf{B}^{2 n} \mathfrak{u}(1). Here b2n𝔲(1)b^{2n} \mathfrak{u}(1) is the abelian 2n-Lie algebra whose underlying complex is concentrated in degree 2n2 n on ℝ\mathbb{R}.

So fully generally, one finds ∞-Lie algebras, ∞-Lie groups and gauge fields behaving like connections with values in these.

Classes of examples

We discuss classes of examples of gauge theories that have been considered. For all of these the configuration space is a space of connections on ∞-bundles{∇}\{\nabla\} over spacetimeXX of sorts, which one might take to be the defining property of a gauge theory. But there are different types of action functionals on these configuration spaces.

In contrast to the ∞\infty-Chern-Simons theory discussed above, the general abstract nature, if any, of the action functional for gravity remains somewhat inconclusive and subject of a plethora of speculations. If one passes from connections to their associated Dirac operators and interprets these as parts of a spectral triple there is the spectral action functional on the space of spectral triples. This we discuss in more detail below.

There are various higher group extensions of the Poincare group and the orthonormal group that lead accordingly to higher order variations of gravity.

Phenomenological models: the standard model and gravity

Theoretical physics consists of two parts: theory and models, laws and initial conditions, axioms and phenomenology.

For instance the theory called general relativity describes the classical dynamics of gravity, but does not predict the value of the cosmological constant. Rather, for each choice of the latter does the theory predict a certain dynamics the large-scale universe.

The theory that describes the fundamental forces and particles except gravity is Yang-Mills theory. This, too, does not predict the fundamental particle species seen in experiments, but for a correct choice and identification of these, the theory does predict the dynamics of these particles, as observed in accelerators.

The total collection of these choices of fundamental particles that are observed in experiments is called the standard model of particle physics. It consists basically of

What precisely the “standard” model of particle physics is changes slightly over time, as new experimental insights are gained. Its particles were added item-by-item as they were discovered. More recently the mass of the particles called neutrinos, which was originally thought to be precisely 0, was measured to be very small, but non-vanishing.

The standard model as far as understood today exhibits a curious mixture of pattern and irregularity. This seems to suggest that it ought to have a more fundamental description in terms of a conceptually simpler structure out of which these patterns with their irregularities emerge. Since also the force of gravity is not presently included in the quantization of the standard model, it may seem plausible that this underlying structure is related to quantum gravity.

We discuss in the following some of the proposals that have been suggested for how to formalize this situation.

This way a single σ\sigma-model may encode a rich multiple particle content and we shall speak of a single superparticle with different excitations or modes .

An early proposal for a single unified connection ∇\nabla that would subsume both gravity as well as Yang-Mills forces in a phenomenologically realistic way is the Kaluza-Klein mechanism. This assumes a single Levi-Civita connection but on a pseudo-Riemannian manifoldXX which is locally of the product form X4×FdX_4 \times F_{d} with X4X_4 a 4-dimensional pseudo-Riemannian manifold and FdF_d a dd-dimensional Riemannian manifold of very small Riemannian volume. As described in more detail at Kaluza-Klein mechanism, this makes the isometry groups of FdF_d appear as extra gauge group factors as seen on X4X_4. As also described in more detail there, while this Ansatz does reproduce the correct general form of gravity coupled to Yang-Mills forces, in its original form it does also have some phenomenologically unviable aspects.

This data is that of spectral triple, which is well known to enocode Riemannian noncommutative geometry (rather: spectral geometry ). It is therefore natural to search for a Kaluza-Klein ansatz in spectral geometry that would produce the standard model context.

Introductions to higher category theory and physics

A historical introduction to some aspects of n-categories (for low nn) in physics can be found here:

John Baez and Aaron Lauda, A prehistory of nn-categorical physics (pdf), to appear in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson.

On formal quantum field theory

FQFT

Kontsevich’s 1994 ICM article, is one of the seminal papers of the 1990s. This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-branes (before physicists realized their role — and directly inspiring many physicists) and the fact that they naturally form dg-categories and A∞A_\infty-category, and thus led to a deluge of papers involving category theory and higher category theory in close relation with mathematical physics.

As far as I can tell, at that time the Yang-Mills terms were stilll included by hand. The main point of the spectral approach was to realize that it could nicely explain the Higgs boson and its Yukawa coupling terms to the fermions. It did (and does) so by realizing the Higgs boson as an internal part of an ordinary minimally coupled connection 1-form - an internal gauge boson.

A few years later Connes apparently realized that also the gravitational and gauge kinetic Yang-Mills terms had an inherent operator-theoretic formulation, namely the spectral action principle.

This is in particular designed to take the ordinary physicst by the hand and introduce him or her gently to the operator-theoretic spectral description by motivating these by the structure of the standard model.

So already over ten years ago people had a pretty good idea that and how the standard model action has an elegant descritption as a spectral action.

The only problem was: this description was wrong. But only in the sense in which ideas in physics tend to be wrong - not entirely wrong but not quite right.

Namely, instead of containing the fermionic particle content of the standard model, these spectral models produced four copies of every fermion in the standard model. A slight overkill.

By a remarkable synchronicity, it seems that there was no progress on this aspect for about ten years, and now two preprints appear almost simultaneously, presenting the solution:

and shown to encode subtle quantum anomaly-cancellation phenomena in gauge theory. A systematic formal description of the nature of these quantum anomalies and further discussion of applications on differential cohomology in physics is in

This article starts with an introduction on basic electromagnetism and points out that already there, in the presence of magnetic charge, a careful analysis of quantum anomalies shows that there are higher gauge theoretic effects even in this familiar theory.

On topos-theoretic formulations of physics

Gros toposes

The observation that any formalization of physics ought to take place in a suitable topos has been promoted early on by Bill Lawvere. Lawvere started out as a student in continuum mechanics and his search for a formalization of physics made him end up being one of the most general abstract thinkers in category theory.

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C-star-algebra of observables AA induces a toposT(A)T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*C^*-algebra. According to the constructiveGelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A)S(A) in T(A)T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*C^*-algebra (which is the noncommutative notion of a space). In this setting, states on AA become probability measures (more precisely, valuations) on S(A)S(A), and self-adjoint elements of AA define continuous functions (more precisely, locale maps) from S(A)S(A) to Scott’s interval domain. Noting that open subsets of S(A)S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by AA is essentially turned into a classical theory, internal to the topos T(A)T(A).

Closely related to this, several approaches at realizing quantum computing rely on TQFT methods and are treated with methods from higher category theory. For instance the notion of a blob n-category orginates in investigations of quantum computing.